Theorem univ 4575

Description: The union of the universe is the universe. Exercise 4.12(c) of [Mendelson] p. 235. (Contributed by NM, 14-Sep-2003.)

Assertion

Ref	Expression
univ	⊢ ∪ V = V

Proof of Theorem univ

Step	Hyp	Ref	Expression
1		pwv 3893	. . 3 ⊢ 𝒫 V = V
2	1	unieqi 3904	. 2 ⊢ ∪ 𝒫 V = ∪ V
3		unipw 4311	. 2 ⊢ ∪ 𝒫 V = V
4	2, 3	eqtr3i 2253	1 ⊢ ∪ V = V

Syntax hints:   = wceq 1397  Vcvv 2801  𝒫 cpw 3653  ∪ cuni 3894

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2204  ax-ext 2212  ax-sep 4208  ax-pow 4266

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1810  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-rex 2515  df-v 2803  df-in 3205  df-ss 3212  df-pw 3655  df-sn 3676  df-uni 3895

This theorem is referenced by: (None)